{"id":65106,"date":"2025-08-18T09:52:59","date_gmt":"2025-08-18T06:52:59","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=65106"},"modified":"2025-08-18T09:52:59","modified_gmt":"2025-08-18T06:52:59","slug":"complex-root","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/complex-root\/","title":{"rendered":"Complex Root Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 class=\"text-lg font-display font-bold\">Calculate the nth Roots of a Complex Number<\/h2><\/div><div class=\"relative group inline-block\">\n  <button class=\"favorite\" id=\"favorite\" data-favorite-id=\"65106\" data-favorite-title=\"Complex Root\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/complex-root\/\" data-favorite-excerpt=\"This calculator finds the &lt;strong&gt;nth roots of complex numbers&lt;\/strong&gt;, showing all possible solutions in both rectangular and polar forms. It\u2019s useful in advanced algebra, complex analysis, and engineering applications where root extraction from complex numbers is required.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-algebra\">\n    <span class=\"icon icon-shape-star-empty\"><\/span>\n  <\/button>\n  <div class=\"absolute right-full -translate-y-1\/2 top-1\/2 mr-2 w-max max-w-xs px-3 py-2 bg-gray-800 text-white text-xs rounded shadow-lg opacity-0 group-hover:opacity-100 transition-opacity duration-200 z-10 pointer-events-none\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/mathematics\/complex-root\/\" method=\"POST\" class=\"calculator\" id=\"calculator-65106\" data-post=\"65106\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Real part (a)<\/label><div class=\"term\">a<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"real_part_a\" id=\"field-1\" step=\"any\" value=\"1\"\/><small>Enter the real part of the complex number<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">Imaginary part (b)<\/label><div class=\"term\">b<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"imaginary_part_b\" id=\"field-2\" step=\"any\" value=\"0\"\/><small>Enter the imaginary part of the complex number<\/small><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Root degree (n)<\/label><div class=\"term\">n<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"root_degree_n\" id=\"field-3\" min=\"1\" step=\"1\" value=\"2\"\/><small>The degree of the root you want to extract<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"65106\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Roots (Rectangular form)<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"65106\" aria-live=\"polite\"><h2 class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/h2><div class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><ins class=\"adsbygoogle\"\n     style=\"display:block; text-align:center; margin: 32px 0;\"\n     data-ad-layout=\"in-article\"\n     data-ad-format=\"fluid\"\n     data-ad-client=\"ca-pub-1721569815777345\"\n     data-ad-slot=\"6317458308\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Complex Root Formula (De Moivre\u2019s Theorem)<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n$$z^{1\/n} = \\sqrt[n]{r} \\left[ \\cos\\left( \\frac{\\theta + 2\\pi k}{n} \\right) + i \\sin\\left( \\frac{\\theta + 2\\pi k}{n} \\right) \\right],\\quad k = 0, 1, \\ldots, n-1$$\n<\/div>\r\n\t<\/figure>\r\n\t<br \/>\nExplanation:<br \/>\nTo find the nth roots of a complex number $$z$$, convert it to polar form $$r(\\cos \\theta + i \\sin \\theta)$$, apply De Moivre\u2019s Theorem, and calculate all $$n$$ distinct roots by rotating $$\\theta$$ around the unit circle.<br \/>\r\n<\/section>\n<section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Complex Root \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p>Find the square roots of z = 1 + i<\/p>\n<p>Step 1: Convert to polar form: r = \u221a2, \u03b8 = \u03c0\/4<br \/>\nStep 2: n = 2 \u2192 calculate two roots<\/p>\n<p>Root 1: \u221a\u221a2 [cos(\u03c0\/8) + i\u00b7sin(\u03c0\/8)]<br \/>\nRoot 2: \u221a\u221a2 [cos(\u03c0\/8 + \u03c0) + i\u00b7sin(\u03c0\/8 + \u03c0)]<\/p>\n<p><strong>Results (approx):<\/strong><br \/>\nRoot 1 \u2248 1.0987 + 0.4551i<br \/>\nRoot 2 \u2248 -1.0987 &#8211; 0.4551i<\/p>\n\r\n<\/section>\n<p>Complex roots are key in solving polynomial equations, analyzing AC circuits, and modeling oscillations. This calculator automates the complex root-finding process, providing both the <strong>modulus\u2013argument form<\/strong> and <strong>standard (a + bi)<\/strong> form, making it easier to interpret and visualize results.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This calculator finds the <strong>nth roots of complex numbers<\/strong>, showing all possible solutions in both rectangular and polar forms. It\u2019s useful in advanced algebra, complex analysis, and engineering applications where root extraction from complex numbers is required.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[11],"class_list":["post-65106","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-algebra"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65106","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=65106"}],"version-history":[{"count":2,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65106\/revisions"}],"predecessor-version":[{"id":65108,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/65106\/revisions\/65108"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=65106"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=65106"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=65106"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}